Question

𝐀𝐏 is the tangent to the circle with centre 𝐎 and radius 𝟖 𝐜𝐦. If 𝐀𝐁 = 𝟗 𝐜𝐦, then the length of the tangent 𝐀𝐏 is?

Answer 1

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

∴ AP ⟂ OP
⇒ ∠OPA = 90° and ΔOPA is a right-angled triangle.
OA = OB + AB = 8 + 9 = 17 cm
Now, applying pythagoras theorem in ΔOPA,
(OP)${}^2$ + (AP)${}^2$ = (OA)${}^2$
(AP)${}^2$ = (OA)${}^2$ − (OP)${}^2$
(AP)${}^2$ = (17)${}^2$ − (8)${}^2$
(AP)${}^2$ = 225
AP = 15 cm